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The Geometry of Niggli Reduction I: The Boundary Polytopes of the Niggli Cone
Correct identification of the Bravais lattice of a crystal is an important
step in structure solution. Niggli reduction is a commonly used technique. We
investigate the boundary polytopes of the Niggli-reduced cone in the
six-dimensional space G6 by algebraic analysis and organized random probing of
regions near 1- through 8-fold boundary polytope intersections. We limit
consideration of boundary polytopes to those avoiding the mathematically
interesting but crystallographically impossible cases of 0 length cell edges.
Combinations of boundary polytopes without a valid intersection in the closure
of the Niggli cone or with an intersection that would force a cell edge to 0 or
without neighboring probe points are eliminated. 216 boundary polytopes are
found: 15 5-D boundary polytopes of the full G6 Niggli cone, 53 4-D boundary
polytopes resulting from intersections of pairs of the 15 5-D boundary
polytopes, 79 3-D boundary polytopes resulting from 2-fold, 3-fold and 4-fold
intersections of the 15 5-D boundary polytopes, 55 2-D boundary polytopes
resulting from 2-fold, 3-fold, 4-fold and higher intersections of the 15 5-D
boundary polytopes, 14 1-D boundary polytopes resulting from 3-fold and higher
intersections of the 15 5-D boundary polytopes. All primitive lattice types can
be represented as combinations of the 15 5-D boundary polytopes. All
non-primitive lattice types can be represented as combinations of the 15 5-D
boundary polytopes and of the 7 special-position subspaces of the 5-D boundary
polytopes. This study provides a new, simpler and arguably more intuitive basis
set for the classification of lattice characters and helps to illuminate some
of the complexities in Bravais lattice identification. The classification is
intended to help in organizing database searches and in understanding which
lattice symmetries are "close" to a given experimentally determined cell
Ehrhart f*-coefficients of polytopal complexes are non-negative integers
The Ehrhart polynomial of an integral polytope counts the number of
integer points in integral dilates of . Ehrhart polynomials of polytopes are
often described in terms of their Ehrhart -vector (aka Ehrhart
-vector), which is the vector of coefficients of with respect to
a certain binomial basis and which coincides with the -vector of a regular
unimodular triangulation of (if one exists). One important result by
Stanley about -vectors of polytopes is that their entries are always
non-negative. However, recent combinatorial applications of Ehrhart theory give
rise to polytopal complexes with -vectors that have negative entries.
In this article we introduce the Ehrhart -vector of polytopes or, more
generally, of polytopal complexes . These are again coefficient vectors of
with respect to a certain binomial basis of the space of polynomials and
they have the property that the -vector of a unimodular simplicial complex
coincides with its -vector. The main result of this article is a counting
interpretation for the -coefficients which implies that -coefficients
of integral polytopal complexes are always non-negative integers. This holds
even if the polytopal complex does not have a unimodular triangulation and if
its -vector does have negative entries. Our main technical tool is a new
partition of the set of lattice points in a simplicial cone into discrete
cones. Further results include a complete characterization of Ehrhart
polynomials of integral partial polytopal complexes and a non-negativity
theorem for the -vectors of rational polytopal complexes.Comment: 19 pages, 1 figur
Labelled tree graphs, Feynman diagrams and disk integrals
In this note, we introduce and study a new class of "half integrands" in
Cachazo-He-Yuan (CHY) formula, which naturally generalize the so-called
Parke-Taylor factors; these are dubbed Cayley functions as each of them
corresponds to a labelled tree graph. The CHY formula with a Cayley function
squared gives a sum of Feynman diagrams, and we represent it by a combinatoric
polytope whose vertices correspond to Feynman diagrams. We provide a simple
graphic rule to derive the polytope from a labelled tree graph, and classify
such polytopes ranging from the associahedron to the permutohedron.
Furthermore, we study the linear space of such half integrands and find (1) a
nice formula reducing any Cayley function to a sum of Parke-Taylor factors in
the Kleiss-Kuijf basis (2) a set of Cayley functions as a new basis of the
space; each element has the remarkable property that its CHY formula with a
given Parke-Taylor factor gives either a single Feynman diagram or zero. We
also briefly discuss applications of Cayley functions and the new basis in
certain disk integrals of superstring theory.Comment: 30+8 pages, many figures;typos fixe
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